Jenga Framework: Multidisciplinary Insights

• Jenga Framework is a multidisciplinary system that unifies combinatorial topology, robotic perception, graph-based modeling, and neural memory strategies.
• It formalizes physical and algorithmic strategies for block selection, extraction, and scene deconstruction, ensuring robust performance in robotics and vision.
• The framework demonstrates practical efficiency improvements in LLM serving and video generation through innovative memory allocation and sparse attention techniques.

The Jenga Framework encompasses a diverse and technically rich set of methodologies unified by the physical and conceptual challenges posed by the game Jenga—strategic block selection, extraction, and the combinatorial or geometric implications of its configurations. Across academic literature, "Jenga Framework" denotes rigorous approaches spanning combinatorial topology, robotic manipulation, scene understanding, memory management for LLMs, and efficient inference pipelines. This entry delineates the foundational geometric analysis, advanced algorithmic strategies, and applied systems engineering captured in the modern Jenga-related literature.

1. Geometric Topology of Jenga-Like Configurations

Jenga frameworks grounded in geometry treat the union of blocks as a polyhedral closed surface, formalizing the analysis using combinatorial topology. The boundary structure is required to satisfy two polyhedral criteria: every edge belongs to exactly two faces, and the link of each vertex is connected. This enables defining the genus $g(Q)$ for the surface $Q$, corresponding to its fundamental topological invariant—the number of holes.

Angular defects at vertices, computed via

$$\kappa(v) = 2\pi - \sum_{f\ni v} \text{angle of } f \text{ at } v,$$

are summed and related to the genus by the Gauss–Bonnet (Descartes’) theorem: $\sum_{v\in V(Q)} \kappa(v) = 4\pi(1-g(Q)).$ Vertex contributions are classified distinctly for each local geometry (Type I, II, III), yielding a explicit genus formula: $$g(Q) = -\frac{N_I}{8} + \frac{N_{II}}{8} + \frac{N_{III}}{4} + 1.$$

Generalizing to $(n,k)$-Jenga games—with $n$ columns and $k$ levels—the maximum genus is realized in specific configurations, governed by whether $n$ is odd or even: $g(n,k) = \begin{cases} \frac{n(n-1)(k-2)}{2} & \text{if %%%%6%%%% odd}, \ \frac{n(n-2)(k-2)}{2} & \text{if %%%%7%%%% even}. \end{cases}$ This reflects a deep synergy between discrete geometry and combinatorial game structure (Akiyama et al., 2017).

2. Robotics: Perception, Planning, and Manipulation

Robotic Jenga frameworks operationalize perception-driven and strategic manipulation. Methods are predicated on high-fidelity instance segmentation (e.g., YOLACT++ with ResNet-50/FPN backbones) for block identification, applying synthetic datasets with controlled variation (lighting, occlusion, material properties). Pixel-wise block masks facilitate accurate pose recovery via the Perspective-n-Points algorithm.

Control strategies integrate real-time visual servoing in eye-in-hand configurations, computing feature error

$e_s = s - s^*,$

with features capturing spatial (x, y), logarithmic depth, and orientation (axis-angle). The robot's velocity command is derived from

$v = -\lambda \cdot L_s^+ \cdot e_s,$

where $L_s^+$ denotes the computed interaction matrix. Tactile feedback, enabled by monodirectional force sensors, introduces force thresholds (e.g., $0.32\,\mathrm{N}$, $0.18\,\mathrm{N}$) for dynamic block removability decision-making.

Experimental systems (e.DO manipulator, Intel RealSense D435i, MicroForce FMA) demonstrate up to 14 consecutive extractions, average spatial errors below 0.2 mm, and robust success rates, underscoring the precision attainable through closed-loop multimodal sensing and control (Marchionna et al., 2022).

3. Strategic Play and Contact-Rich Manipulation via Graph-Based Modeling

Frameworks utilizing graph-based models formalize block selection and extraction as linked inference and prediction problems. The tower state is abstracted as a graph $G_s = (V, E)$, with nodes encoding block geometry, weight, and candidate status. Edges capture direct physical support relations.

Block selection employs graph convolutional networks (GCNs), achieving 74% accuracy in simulation (Isaac Sim), outperforming pose-augmented graphs. Block extraction is addressed using graph network-based simulators, predicting block displacements $(\Delta\hat{V})$ under candidate actions $a_t$. Model predictive path integral (MPPI) control loops sample actions and evaluate

$$O(a_t, b, V_t, V_{t+1}) = O_a(a_t) + O_d(b, V_t, V_{t+1}),$$

where $O_a(a_t)$ rewards extraction-aligned movement and penalizes lateral displacement, and $O_d$ penalizes spurious block movements. Extraction success rates of 65% (enhanced by dynamics-aware penalties) are reported, with these strategic and reactive methods offering generalizability for broader contact-rich multi-object manipulation (Puthuveetil et al., 14 May 2025).

4. Jenga Frameworks for Vision-Based Grasping in Structured Stacks

Addressing grasping from stacks, this class of Jenga frameworks fuses 2D detection (YOLOv8), 6DoF pose estimation (RGB ZebraPose), and dual filtering mechanisms. Each candidate undergoes a calculation of visibility ratio

$r^i = |M_m^i| / |M_a^i|,$

where $M_m^i$ is the modal mask and $M_a^i$ is the rendered amodal mask; failure to meet threshold $\epsilon_{vis}$ disqualifies truncated candidates. IMU-based height filtering exploits the gravity-projected vertical component,

$o_g^{(i,z)} = \alpha_s \cdot o_s^i,$

to prioritize unobstructed, surface-level objects.

Evaluation on synthetic BlenderProc datasets (ground truth with pose, segmentation, graspability) employs precision/AP metrics based on ADD-S and MSSD thresholds. Filtering yields mean AP up to 0.84, far surpassing baseline pose-only hypotheses. Real-world deployments validate performance, but highlight the challenge of error-free operation, particularly under occlusion and ambiguous height registration (Jeevanandam et al., 16 Jun 2025).

5. Scene Understanding: Visual Jenga and Counterfactual Inpainting

The Visual Jenga task extends the conceptual foundation of Jenga to scene understanding by sequentially removing objects from an image while maintaining physical and geometric scene coherence. The method leverages asymmetry in object dependency: objects are ranked via the diversity of plausible inpaintings produced by large generative models upon their removal.

For an object $A$ in image $X$, inpainting the region yields samples $\left\{c_{\text{new}}^j\right\}_{j=1}^N$, and the diversity score

$$1 - \left[ \left(\frac{1}{N} \sum_{j=1}^N \text{ClipSim}(c_{\text{new}}^j, c_{\text{orig}})\right) \cdot \left(\frac{1}{N} \sum_{j=1}^N \text{DinoSim}(c_{\text{new}}^j, c_{\text{orig}})\right) \right]$$

quantifies object "replaceability". High accuracy (91% on NYU-v2, 70% on HardParse) is reported for support relations, outperforming simple ordering heuristics (Bhattad et al., 27 Mar 2025).

A plausible implication is the methodology's utility for robotics (object selection/sequencing), image editing (scene coherence), and synthetic scene generation, especially where structured dependencies must be preserved.

6. Computational Efficiency in LLM Serving and Video Generation

The Jenga Framework also denotes memory management and inference strategies in modern neural systems, notably LLM serving and video diffusion models.

For LLMs with heterogeneous embeddings, Jenga introduces a two-level allocator, leveraging compatible page sizes set by the least common multiple (LCM) of all sizes, e.g.,

$$\text{Page size} = \mathop{\mathrm{LCM}}(256,384)=768~\text{bytes}$$

for 256- and 384-byte embeddings. Layer-specific APIs (e.g., update_last_access) allow flexible caching policies, adapting to varying token dependencies (full-prefix, sliding window, last-token). Implemented on vLLM, Jenga yields up to 79.6% GPU memory utilization improvement and up to 4.92$\times$ throughput speedup (Zhang et al., 24 Mar 2025).

For video generation, Jenga applies block-wise sparse attention via space-filling curves and progressive resolution generation—starting inference at low latent resolutions and upsampling in stages with re-noising. Block relevance scores

$$\mathcal{R} = \text{softmax}(\hat{Q}\hat{K}^\top/\sqrt{d_k})$$

and selective computation reduce quadratic complexity. Empirical evaluations on state-of-the-art DiTs show speedups up to 8.83$\times$ with negligible quality loss ($<0.01\%$ on VBench) (Zhang et al., 22 May 2025). The plug-and-play design and open-source release advance practical deployment on large hardware.

7. Synthesis and Research Landscape

The Jenga Framework, in its many incarnations, serves as a model for translating combinatorial, geometric, and dynamical game-theoretic challenges into formal systems—spanning surface topology, robotic manipulation, vision-driven grasping, scene deconstruction, high-efficiency LLM memory management, and scalable video generation. Each instantiation is characterized by rigorous formulation, explicit metrics, and practical (often experimentally validated) strategies.

These frameworks collectively illuminate structural dependency, connectivity, and efficiency in complex multi-object, multi-agent, or multi-modal systems. The technical lineage—from angular defect analysis and Gauss–Bonnet applications to graph neural networks and memory allocation—reflects a multifaceted research trajectory driven by the foundational constraints and opportunities epitomized by Jenga.